Pronilpotent quotients associated with primitive substitutions

Goulet-Ouellet, Herman

doi:10.1016/j.jalgebra.2022.05.021

Cited by 2 publications

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References 13 publications

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“…The behavior of return sets in a given shift space has also been linked, by Almeida and Costa, to the Schützenberger group; in particular, global stability of the subgroups generated by the return sets entails freeness of the Schützenberger group [3]. Together with results from [12,18], this can be used to show that dendric shift spaces are not topologically conjugate to shift spaces of primitive uniform substitutions-though the same conclusion can also be reached using other means [9].…”

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On Substitutions Preserving Their Return Sets

Berthé

Goulet-Ouellet

2023

Lecture Notes in Computer Science

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We consider the question of whether or not a given primitive substitution preserves its sets of return words-or return sets for short. More precisely, we study the property asking that the image of the return set to a word equals the return set to the image of that word. We show that, for bifix encodings (where images of letters form a bifix code), this property holds for all but finitely many words. On the other hand, we also show that every conjugacy class of Sturmian substitutions contains a member for which the property fails infinitely often. Various applications and examples of these results are presented, including a description of the subgroups generated by the return sets in the shift of the Thue-Morse substitution. Up to conjugacy, these subgroups can be sorted into strictly decreasing chains of isomorphic subgroups weaving together a simple pattern. This is in stark contrast with the Sturmian case, and more generally with the dendric case (including in particular the Arnoux-Rauzy case), where it is known that all return sets generate the free group over the underlying alphabet.

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On Substitutions Preserving Their Return Sets

Berthé

Goulet-Ouellet

2023

Lecture Notes in Computer Science

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A profinite approach to complete bifix decodings of recurrent languages

Costa

2023

Forum Mathematicum

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We approach the study of complete bifix decodings of (uniformly) recurrent languages with the help of free profinite monoids. We show that the complete bifix decoding of a uniformly recurrent language F by an F-charged rational complete bifix code is uniformly recurrent. An analogous result is obtained for recurrent languages. As an application of the machinery developed within this approach, we show that the maximal pronilpotent quotient of the Schützenberger group of an irreducible symbolic dynamical system is an invariant of eventual conjugacy.

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Pronilpotent quotients associated with primitive substitutions

Cited by 2 publications

References 13 publications

On Substitutions Preserving Their Return Sets

On Substitutions Preserving Their Return Sets

A profinite approach to complete bifix decodings of recurrent languages

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